We consider high-dimensional quadratic classifiers in non-sparse settings. The target of classification rules is not Bayes error rates in the context. The classifier based on the Mahalanobis distance does not always give a preferable performance even if the populations are normal distributions having known covariance matrices. The quadratic classifiers proposed in this paper draw information about heterogeneity effectively through both the differences of expanding mean vectors and covariance matrices. We show that they hold a consistency property in which misclassification rates tend to zero as the dimension goes to infinity under non-sparse settings. We verify that they are asymptotically distributed as a normal distribution under certain conditions. We also propose a quadratic classifier after feature selection by using both the differences of mean vectors and covariance matrices. Finally, we discuss performances of the classifiers in actual data analyses. The proposed classifiers achieve highly accurate classification with very low computational costs.
We consider high-dimensional multiclass classification by sparse multinomial logistic regression. Unlike binary classification, in the multiclass setup one can think about an entire spectrum of possible notions of sparsity associated with different structural assumptions on the regression coefficients matrix. We propose a computationally feasible feature selection procedure based on penalized maximum likelihood with convex penalties capturing a specific type of sparsity at hand. In particular, we consider global sparsity, double row-wise sparsity, and low-rank sparsity, and show that with the …
The nearest-centroid classifier is a simple linear-time classifier based on computing the centroids of the data classes in the training phase, and then assigning a new datum to the class corresponding to its nearest centroid. Thanks to its very low computational cost, the nearest-centroid classifier is still widely used in machine learning, despite the development of many other more sophisticated classification methods. In this paper, we propose two sparse variants of the nearest-centroid classifier, based respectively on $\ell_1$ and $\ell_2$ distance criteria. The proposed sparse classifiers…
AUC (Area under the ROC curve) is an important performance measure for applications where the data is highly imbalanced. Learning to maximize AUC performance is thus an important research problem. Using a max-margin based surrogate loss function, AUC optimization problem can be approximated as a pairwise rankSVM learning problem. Batch learning methods for solving the kernelized version of this problem suffer from scalability and may not result in sparse classifiers. Recent years have witnessed an increased interest in the development of online or single-pass online learning algorithms that de…
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；Duke University（杜克大学）
The use of quadratic discriminant analysis (QDA) or its regularized version (R-QDA) for classification is often not recommended, due to its well-acknowledged high sensitivity to the estimation noise of the covariance matrix. This becomes all the more the case in unbalanced data settings for which it has been found that R-QDA becomes equivalent to the classifier that assigns all observations to the same class. In this paper, we propose an improved R-QDA that is based on the use of two regularization parameters and a modified bias, properly chosen to avoid inappropriate behaviors of R-QDA in unb…
Quadratic discriminant analysis (QDA) is a widely used classification technique that generalizes the linear discriminant analysis (LDA) classifier to the case of distinct covariance matrices among classes. For the QDA classifier to yield high classification performance, an accurate estimation of the covariance matrices is required. Such a task becomes all the more challenging in high dimensional settings, wherein the number of observations is comparable with the feature dimension. A popular way to enhance the performance of QDA classifier under these circumstances is to regularize the covarian…
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Swarthmore College（斯沃斯莫尔学院）；Villanova University（维拉诺瓦大学）
Although deep learning has shown great success in recent years, researchers have discovered a critical flaw where small, imperceptible changes in the input to the system can drastically change the output classification. These attacks are exploitable in nearly all of the existing deep learning classification frameworks. However, the susceptibility of deep sparse coding models to adversarial examples has not been examined. Here, we show that classifiers based on a deep sparse coding model whose classification accuracy is competitive with a variety of deep neural network models are robust to adve…
We develop a class of rules spanning the range between quadratic discriminant analysis and naive Bayes, through a path of sparse graphical models. A group lasso penalty is used to introduce shrinkage and encourage a similar pattern of sparsity across precision matrices. It gives sparse estimates of interactions and produces interpretable models. Inspired by the connected-components structure of the estimated precision matrices, we propose the community Bayes model, which partitions features into several conditional independent communities and splits the classification problem into separate sma…
Yahoo! Labs（雅虎实验室）；Indian Institute of Science（印度科学学院）
We consider the problem of designing a sparse Gaussian process classifier (SGPC) that generalizes well. Viewing SGPC design as constructing an additive model like in boosting, we present an efficient and effective SGPC design method to perform a stage-wise optimization of a predictive loss function. We introduce new methods for two key components viz., site parameter estimation and basis vector selection in any SGPC design. The proposed adaptive sampling based basis vector selection method aids in achieving improved generalization performance at a reduced computational cost. This method can al…
Quadratic regression involves modeling the response as a (generalized) linear function of not only the features $x^{j_1}$ but also of quadratic terms $x^{j_1}x^{j_2}$. The inclusion of such higher-order "interaction terms" in regression often provides an easy way to increase accuracy in already-high-dimensional problems. However, this explodes the problem dimension from linear $O(p)$ to quadratic $O(p^2)$, and it is common to look for sparse interactions (typically via heuristics). In this paper, we provide a new algorithm - Interaction Hard Thresholding (IntHT) which is the first one to prova…
We consider the problem of the recovery of a k-sparse vector from compressed linear measurements when data are corrupted by a quantization noise. When the number of measurements is not sufficiently large, different $k$-sparse solutions may be present in the feasible set, and the classical l1 approach may be unsuccessful. For this motivation, we propose a non-convex quadratic programming method, which exploits prior information on the magnitude of the non-zero parameters. This results in a more efficient support recovery. We provide sufficient conditions for successful recovery and numerical si…
University of Paris-Sud（巴黎第十一大学）；CNRS（法国国家科学研究中心）；Hungarian Academy of Sciences（匈牙利科学院）；University of Szeged（塞格德大学）
In this paper we propose an algorithm that builds sparse decision DAGs (directed acyclic graphs) from a list of base classifiers provided by an external learning method such as AdaBoost. The basic idea is to cast the DAG design task as a Markov decision process. Each instance can decide to use or to skip each base classifier, based on the current state of the classifier being built. The result is a sparse decision DAG where the base classifiers are selected in a data-dependent way. The method has a single hyperparameter with a clear semantics of controlling the accuracy/speed trade-off. The al…
The study of healthy brain development helps to better understand the brain transformation and brain connectivity patterns which happen during childhood to adulthood. This study presents a sparse machine learning solution across whole-brain functional connectivity (FC) measures of three sets of data, derived from resting state functional magnetic resonance imaging (rs-fMRI) and task fMRI data, including a working memory n-back task (nb-fMRI) and an emotion identification task (em-fMRI). These multi-modal image data are collected on a sample of adolescents from the Philadelphia Neurodevelopment…
Ben-Gurion University of the Negev（内盖夫本-古里安大学）；Benin School of Computer Science and Engineering（贝宁计算机科学与工程学院）；The Hebrew University（希伯来大学）；Toyota Technological Institute at Chicago（芝加哥丰田技术研究院）；Columbia University（哥伦比亚大学）；Rutgers University（罗格斯大学）
We consider the problem of learning a non-negative linear classifier with a $1$-norm of at most $k$, and a fixed threshold, under the hinge-loss. This problem generalizes the problem of learning a $k$-monotone disjunction. We prove that we can learn efficiently in this setting, at a rate which is linear in both $k$ and the size of the threshold, and that this is the best possible rate. We provide an efficient online learning algorithm that achieves the optimal rate, and show that in the batch case, empirical risk minimization achieves this rate as well. The rates we show are tighter than the u…
Indian Statistical Institute（印度统计研究所）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Quadratic discriminant analysis (QDA) is a widely used statistical tool to classify observations from different multivariate Normal populations. The generalized quadratic discriminant analysis (GQDA) classification rule/classifier, which generalizes the QDA and the minimum Mahalanobis distance (MMD) classifiers to discriminate between populations with underlying elliptically symmetric distributions competes quite favorably with the QDA classifier when it is optimal and performs much better when QDA fails under non-Normal underlying distributions, e.g. Cauchy distribution. However, the classifi…
IBM Research（IBM研究院）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
We consider support recovery in the quadratic logistic regression setting - where the target depends on both p linear terms $x_i$ and up to $p^2$ quadratic terms $x_i x_j$. Quadratic terms enable prediction/modeling of higher-order effects between features and the target, but when incorporated naively may involve solving a very large regression problem. We consider the sparse case, where at most $s$ terms (linear or quadratic) are non-zero, and provide a new faster algorithm. It involves (a) identifying the weak support (i.e. all relevant variables) and (b) standard logistic regression optimiz…
University of Minnesota（明尼苏达大学）；Johns Hopkins University（约翰斯·霍普金斯大学）；Princeton University（普林斯顿大学）；Georgia Institute of Technology（佐治亚理工学院）
We propose a DC proximal Newton algorithm for solving nonconvex regularized sparse learning problems in high dimensions. Our proposed algorithm integrates the proximal Newton algorithm with multi-stage convex relaxation based on the difference of convex (DC) programming, and enjoys both strong computational and statistical guarantees. Specifically, by leveraging a sophisticated characterization of sparse modeling structures/assumptions (i.e., local restricted strong convexity and Hessian smoothness), we prove that within each stage of convex relaxation, our proposed algorithm achieves (local) …
We consider the problem of high-dimensional classification between the two groups with unequal covariance matrices. Rather than estimating the full quadratic discriminant rule, we propose to perform simultaneous variable selection and linear dimension reduction on original data, with the subsequent application of quadratic discriminant analysis on the reduced space. In contrast to quadratic discriminant analysis, the proposed framework doesn't require estimation of precision matrices and scales linearly with the number of measurements, making it especially attractive for the use on high-dimens…
The L1-regularized Gaussian maximum likelihood estimator (MLE) has been shown to have strong statistical guarantees in recovering a sparse inverse covariance matrix, or alternatively the underlying graph structure of a Gaussian Markov Random Field, from very limited samples. We propose a novel algorithm for solving the resulting optimization problem which is a regularized log-determinant program. In contrast to recent state-of-the-art methods that largely use first order gradient information, our algorithm is based on Newton's method and employs a quadratic approximation, but with some modific…
Delft University of Technology（代尔夫特理工大学）；Stanford University（斯坦福大学）；VinAI Research（VinAI研究院）；Umea University（于默奥大学）
We propose a novel approximation hierarchy for cardinality-constrained, convex quadratic programs that exploits the rank-dominating eigenvectors of the quadratic matrix. Each level of approximation admits a min-max characterization whose objective function can be optimized over the binary variables analytically, while preserving convexity in the continuous variables. Exploiting this property, we propose two scalable optimization algorithms, coined as the "best response" and the "dual program", that can efficiently screen the potential indices of the nonzero elements of the original program. We…
Tel Aviv University（特拉维夫大学）；The Open University of Israel（以色列开放大学）
In this paper we consider high-dimensional multiclass classification by sparse multinomial logistic regression. We propose first a feature selection procedure based on penalized maximum likelihood with a complexity penalty on the model size and derive the nonasymptotic bounds for misclassification excess risk of the resulting classifier. We establish also their tightness by deriving the corresponding minimax lower bounds. In particular, we show that there exist two regimes corresponding to small and large number of classes. The bounds can be reduced under the additional low noise condition. To…
Syracuse University（雪城大学）；University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）
In the problem of learning a mixture of linear classifiers, the aim is to learn a collection of hyperplanes from a sequence of binary responses. Each response is a result of querying with a vector and indicates the side of a randomly chosen hyperplane from the collection the query vector belongs to. This model provides a rich representation of heterogeneous data with categorical labels and has only been studied in some special settings. We look at a hitherto unstudied problem of query complexity upper bound of recovering all the hyperplanes, especially for the case when the hyperplanes are spa…
Institute of Statistical Science, Academia Sinica（中央研究院统计科学研究所）
The model interpretation is essential in many application scenarios and to build a classification model with a ease of model interpretation may provide useful information for further studies and improvement. It is common to encounter with a lengthy set of variables in modern data analysis, especially when data are collected in some automatic ways. This kinds of datasets may not collected with a specific analysis target and usually contains redundant features, which have no contribution to a the current analysis task of interest. Variable selection is a common way to increase the ability of mod…
Ecole Polytechnique Federale de Lausanne (EPFL)（洛桑联邦理工学院）；The University of Edinburgh（爱丁堡大学）
Classifiers based on sparse representations have recently been shown to provide excellent results in many visual recognition and classification tasks. However, the high cost of computing sparse representations at test time is a major obstacle that limits the applicability of these methods in large-scale problems, or in scenarios where computational power is restricted. We consider in this paper a simple yet efficient alternative to sparse coding for feature extraction. We study a classification scheme that applies the soft-thresholding nonlinear mapping in a dictionary, followed by a linear cl…
The covariance matrix of a $p$-dimensional random variable is a fundamental quantity in data analysis. Given $n$ i.i.d. observations, it is typically estimated by the sample covariance matrix, at a computational cost of $O(np^{2})$ operations. When $n,p$ are large, this computation may be prohibitively slow. Moreover, in several contemporary applications, the population matrix is approximately sparse, and only its few large entries are of interest. This raises the following question, at the focus of our work: Assuming approximate sparsity of the covariance matrix, can its large entries be dete…
Graduate School of Keio University（庆应义塾大学研究生院）；Keio University（庆应义塾大学）
Machine learning has played an important role in information retrieval (IR) in recent times. In search engines, for example, query keywords are accepted and documents are returned in order of relevance to the given query; this can be cast as a multi-label ranking problem in machine learning. Generally, the number of candidate documents is extremely large (from several thousand to several million); thus, the classifier must handle many labels. This problem is referred to as extreme multi-label classification (XMLC). In this paper, we propose a novel approach to XMLC termed the Sparse Weighted N…
RIKEN（理化学研究所）；Instituto Argentino de Radioastronomía, CONICET-CCT La Plata / CIC-PBA / UNLP（阿根廷射电天文研究所，CONICET-拉普拉塔科技中心 / 布宜诺斯艾利斯省科学调查委员会 / 拉普拉塔国立大学）；Southeast University（东南大学）；University of Vic - Central University of Catalonia（维克大学-加泰罗尼亚中央大学）
In this paper, the problem of training a classifier on a dataset with incomplete features is addressed. We assume that different subsets of features (random or structured) are available at each data instance. This situation typically occurs in the applications when not all the features are collected for every data sample. A new supervised learning method is developed to train a general classifier, such as a logistic regression or a deep neural network, using only a subset of features per sample, while assuming sparse representations of data vectors on an unknown dictionary. Sufficient conditio…
We consider the two-group classification problem and propose a kernel classifier based on the optimal scoring framework. Unlike previous approaches, we provide theoretical guarantees on the expected risk consistency of the method. We also allow for feature selection by imposing structured sparsity using weighted kernels. We propose fully-automated methods for selection of all tuning parameters, and in particular adapt kernel shrinkage ideas for ridge parameter selection. Numerical studies demonstrate the superior classification performance of the proposed approach compared to existing nonparam…
ETH Zürich（苏黎世联邦理工学院）；ETH AI Center（苏黎世联邦理工学院人工智能中心）
Popular iterative algorithms such as boosting methods and coordinate descent on linear models converge to the maximum $\ell_1$-margin classifier, a.k.a. sparse hard-margin SVM, in high dimensional regimes where the data is linearly separable. Previous works consistently show that many estimators relying on the $\ell_1$-norm achieve improved statistical rates for hard sparse ground truths. We show that surprisingly, this adaptivity does not apply to the maximum $\ell_1$-margin classifier for a standard discriminative setting. In particular, for the noiseless setting, we prove tight upper and lo…
We consider a discrete optimization formulation for learning sparse classifiers, where the outcome depends upon a linear combination of a small subset of features. Recent work has shown that mixed integer programming (MIP) can be used to solve (to optimality) $\ell_0$-regularized regression problems at scales much larger than what was conventionally considered possible. Despite their usefulness, MIP-based global optimization approaches are significantly slower compared to the relatively mature algorithms for $\ell_1$-regularization and heuristics for nonconvex regularized problems. We aim to b…
TU Munich（慕尼黑工业大学）；University of Bonn（波恩大学）；Halmstad University（哈尔姆斯塔德大学）
We address the non-convex optimisation problem of finding a sparse matrix on the Stiefel manifold (matrices with mutually orthogonal columns of unit length) that maximises (or minimises) a quadratic objective function. Optimisation problems on the Stiefel manifold occur for example in spectral relaxations of various combinatorial problems, such as graph matching, clustering, or permutation synchronisation. Although sparsity is a desirable property in such settings, it is mostly neglected in spectral formulations since existing solvers, e.g. based on eigenvalue decomposition, are unable to acco…
Quadratic discriminant analysis (QDA) is a standard tool for classification due to its simplicity and flexibility. Because the number of its parameters scales quadratically with the number of the variables, QDA is not practical, however, when the dimensionality is relatively large. To address this, we propose a novel procedure named DA-QDA for QDA in analyzing high-dimensional data. Formulated in a simple and coherent framework, DA-QDA aims to directly estimate the key quantities in the Bayes discriminant function including quadratic interactions and a linear index of the variables for classif…
Michigan State University（密歇根州立大学）；GE Global Research（通用电气全球研究中心）
In this paper, we study the problem of sparse multiple kernel learning (MKL), where the goal is to efficiently learn a combination of a fixed small number of kernels from a large pool that could lead to a kernel classifier with a small prediction error. We develop an efficient algorithm based on the greedy coordinate descent algorithm, that is able to achieve a geometric convergence rate under appropriate conditions. The convergence rate is achieved by measuring the size of functional gradients by an empirical $\ell_2$ norm that depends on the empirical data distribution. This is in contrast t…
High-Dimensional Analysis of Single-Layer Attention for Sparse-Token Classification
Department of Statistics, Harvard University（哈佛大学统计学系）；Center of Mathematical Sciences and Applications, Harvard University（哈佛大学数学科学中心）；John A. Paulson School of Engineering and Applied Sciences, Harvard University（哈佛大学工程与应用科学学院）
In this paper we propose a novel method for semi-supervised anomaly detection (SSAD). Our classifier is named QMS22 as its inception was dated 2022 upon the framework of quadratic multiform separation (QMS), a recently introduced classification model. QMS22 tackles SSAD by solving a multi-class classification problem involving both the training set and the test set of the original problem. The classification problem intentionally includes classes with overlapping samples. One of the classes contains mixture of normal samples and outliers, and all other classes contain only normal samples. An o…
Optimisation and Logistics Group, The University of Adelaide（阿德莱德大学优化与物流团队）
A classification algorithm, called the Linear Centralization Classifier (LCC), is introduced. The algorithm seeks to find a transformation that best maps instances from the feature space to a space where they concentrate towards the center of their own classes, while maximimizing the distance between class centers. We formulate the classifier as a quadratic program with quadratic constraints. We then simplify this formulation to a linear program that can be solved effectively using a linear programming solver (e.g., simplex-dual). We extend the formulation for LCC to enable the use of kernel f…
University of Zurich（苏黎世大学）；Université de Nice Sofia Antipolis（尼斯索菲亚安蒂波利斯大学）；Université de Bretagne Sud（南布列塔尼大学）
In this paper, we tackle the question of discovering an effective set of spatial filters to solve hyperspectral classification problems. Instead of fixing a priori the filters and their parameters using expert knowledge, we let the model find them within random draws in the (possibly infinite) space of possible filters. We define an active set feature learner that includes in the model only features that improve the classifier. To this end, we consider a fast and linear classifier, multiclass logistic classification, and show that with a good representation (the filters discovered), such a sim…
University of Cambridge（剑桥大学）；Faculty of Computing, Union University, Belgrade（贝尔格莱德联盟大学计算学院）；University of Amsterdam（阿姆斯特丹大学）
Recent advances in representation learning on graphs, mainly leveraging graph convolutional networks, have brought a substantial improvement on many graph-based benchmark tasks. While novel approaches to learning node embeddings are highly suitable for node classification and link prediction, their application to graph classification (predicting a single label for the entire graph) remains mostly rudimentary, typically using a single global pooling step to aggregate node features or a hand-designed, fixed heuristic for hierarchical coarsening of the graph structure. An important step towards a…
In this paper, we propose a semi-supervised dictionary learning method that uses both the information in labelled and unlabelled data and jointly trains a linear classifier embedded on the sparse codes. The manifold structure of the data in the sparse code space is preserved using the same approach as the Locally Linear Embedding method (LLE). This enables one to enforce the predictive power of the unlabelled data sparse codes. We show that our approach provides significant improvements over other methods. The results can be further improved by training a simple nonlinear classifier as SVM on …
In many compressive sensing problems today, the relationship between the measurements and the unknowns could be nonlinear. Traditional treatment of such nonlinear relationships have been to approximate the nonlinearity via a linear model and the subsequent un-modeled dynamics as noise. The ability to more accurately characterize nonlinear models has the potential to improve the results in both existing compressive sensing applications and those where a linear approximation does not suffice, e.g., phase retrieval. In this paper, we extend the classical compressive sensing framework to a second-…
Recently, techniques have been developed to provably guarantee the robustness of a classifier to adversarial perturbations of bounded L_1 and L_2 magnitudes by using randomized smoothing: the robust classification is a consensus of base classifications on randomly noised samples where the noise is additive. In this paper, we extend this technique to the L_0 threat model. We propose an efficient and certifiably robust defense against sparse adversarial attacks by randomly ablating input features, rather than using additive noise. Experimentally, on MNIST, we can certify the classifications of o…
Regularized discriminant analysis (RDA), proposed by Friedman (1989), is a widely popular classifier that lacks interpretability and is impractical for high-dimensional data sets. Here, we present an interpretable and computationally efficient classifier called high-dimensional RDA (HDRDA), designed for the small-sample, high-dimensional setting. For HDRDA, we show that each training observation, regardless of class, contributes to the class covariance matrix, resulting in an interpretable estimator that borrows from the pooled sample covariance matrix. Moreover, we show that HDRDA is equivale…
The Johns Hopkins University（约翰斯·霍普金斯大学）；U.S. Army Research Laboratory（美国陆军研究实验室）
Pixel-wise classification, where each pixel is assigned to a predefined class, is one of the most important procedures in hyperspectral image (HSI) analysis. By representing a test pixel as a linear combination of a small subset of labeled pixels, a sparse representation classifier (SRC) gives rather plausible results compared with that of traditional classifiers such as the support vector machine (SVM). Recently, by incorporating additional structured sparsity priors, the second generation SRCs have appeared in the literature and are reported to further improve the performance of HSI. These p…
Optimality of Message-Passing Architectures for Sparse Graphs
David R. Cheriton School of Computer Science, University of Waterloo（滑铁卢大学戴维·R·切里顿计算机科学学院）；Department of Applied Mathematics, University of Waterloo（滑铁卢大学应用数学系）
Weizmann Institute of Science（魏茨曼科学研究所）；Princeton University（普林斯顿大学）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Texas A&M University（得克萨斯农工大学）
Several classification methods assume that the underlying distributions follow tree-structured graphical models. Indeed, trees capture statistical dependencies between pairs of variables, which may be crucial to attain low classification errors. The resulting classifier is linear in the log-transformed univariate and bivariate densities that correspond to the tree edges. In practice, however, observed data may not be well approximated by trees. Yet, motivated by the importance of pairwise dependencies for accurate classification, here we propose to approximate the optimal decision boundary by …
Laboratoire Hubert Curien UMR CNRS 5516（于贝尔·居里安实验室（法国国家科学研究中心 UMR 5516））
In recent years, the crucial importance of metrics in machine learning algorithms has led to an increasing interest for optimizing distance and similarity functions. Most of the state of the art focus on learning Mahalanobis distances (requiring to fulfill a constraint of positive semi-definiteness) for use in a local k-NN algorithm. However, no theoretical link is established between the learned metrics and their performance in classification. In this paper, we make use of the formal framework of good similarities introduced by Balcan et al. to design an algorithm for learning a non PSD linea…
